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PreprintCC BY 4.0arXiv

On the Exact Turán Number of F-4,3

  • Chun-Qiu Fang
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Abstract

For a $3$-graph F, the Turán number of F, denoted by ex(n,F), is the maximum number of edges in a $3$-graph on n vertices containing no subgraph isomorphic to F. Let F-4,3 be the $3$-graph formed by a complete four-vertex core and three outer vertices, with all but one of the twelve triples containing one core vertex and two outer vertices. We prove that, for every n≥8, ex(n,F-4,3)=binom n3−binom⌊ n/2⌋3−binom⌈ n/2⌉3, and the balanced complete bipartite $3$-graph is the unique extremal configuration. This determines the exact value and all equality cases in the asymptotic theorem of Mubayi and Rödl. It also extends the exact Turán Number of F3,3 and resolves a conjecture of Frankl, Huang and Rödl.

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Cite

BibTeXRIS
BibTeX
@misc{fang2026exact,
  title = {{On the Exact Turán Number of $F^-_4,3$}},
  author = {Chun-Qiu Fang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29903},
  eprint = {2609.29903},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29903},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the Exact Turán Number of F^-_4,3
AU  - Chun-Qiu Fang
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.29903
UR  - https://arxiv.org/abs/2609.29903
AB  - For a $3$-graph F, the Turán number of F, denoted by ex(n,F), is the maximum number of edges in a $3$-graph on n vertices containing no subgraph isomorphic to F. Let F^-_4,3 be the $3$-graph formed by a complete four-vertex core and three outer vertices, with all but one of the twelve triples containing one core vertex and two outer vertices. We prove that, for every n≥8, ex(n,F^-_4,3)=binom n3−binom⌊ n/2⌋3−binom⌈ n/2⌉3, and the balanced complete bipartite $3$-graph is the unique extremal configuration. This determines the exact value and all equality cases in the asymptotic theorem of Mubayi and Rödl. It also extends the exact Turán Number of F_3,3 and resolves a conjecture of Frankl, Huang and Rödl.
ER  -